Math 2.0

Relativity and gravitational theory Definition

Vacuum solution

Builds on Lorentzian manifoldRicci curvatureEinstein tensorEinstein field equations

Definition

Let (M,g)(M, g) be a 4-dimensional Lorentzian manifold. It is a vacuum solution if its Einstein tensor vanishes: Gab=Rab−12R gab=0,G_{ab} = R_{ab} - \tfrac12 R \, g_{ab} = 0, where RabR_{ab} is the Ricci tensor and RR the scalar curvature of gg.

Remarks

Taking the trace gives R−2R=−R=0R - 2R = -R = 0, so the condition is equivalent to Rab=0R_{ab} = 0: vacuum solutions are the Ricci-flat Lorentzian 4-manifolds. They are the solutions of the Einstein field equations with Tab=0T_{ab} = 0 and Λ=0\Lambda = 0. Example: the Schwarzschild metric on r>2Mr > 2M has Rab=0R_{ab} = 0, although its Riemann tensor is not zero. Non-example: the Reissner–Nordström metric with charge Q≠0Q \ne 0 is not a vacuum solution, since its electromagnetic field has a nonzero stress–energy tensor and so Rab≠0R_{ab} \ne 0. When a cosmological constant is present, the analogous condition Rab=ΛgabR_{ab} = \Lambda g_{ab} defines the lambdavacuum solutions, such as de Sitter space. For a vacuum solution the Riemann tensor equals the Weyl tensor.

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Einstein tensor, concept, no evidence levelEinstein tensorconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptMatrix, definition, no evidence levelMatrixdefinitionPseudo-Riemannian manifold, concept, no evidence levelPseudo-Riemannian man…conceptRicci curvature, concept, no evidence levelRicci curvatureconceptRiemann curvature tensor, concept, no evidence levelRiemann curvature ten…conceptScalar curvature, concept, no evidence levelScalar curvatureconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptTensor field, concept, no evidence levelTensor fieldconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionEinstein field equations, definition, no evidence levelEinstein field equati…definitionLorentzian manifold, definition, no evidence levelLorentzian manifolddefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionRicci curvature, definition, no evidence levelRicci curvaturedefinitionVacuum solution, definition, no evidence levelVacuum solutiondefinition
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  1. Einstein tensor concept
  2. Lorentzian manifold concept
  3. Matrix definition
  4. Pseudo-Riemannian manifold concept
  5. Ricci curvature concept
  6. Riemann curvature tensor concept
  7. Scalar curvature concept
  8. Signature of a quadratic form concept
  9. Tensor field concept
  10. Square matrix definition
  11. Einstein field equations definition
  12. Lorentzian manifold definition
  13. Trace of a matrix definition
  14. Ricci curvature definition
  15. Vacuum solution definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Vacuum_solution
About this page
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URIm2://314H8S0566
Kinddefinition
WritingW0 stub
MSC83C15, 83C05
DefinesVacuum solution
UsesLorentzian manifold, Ricci curvature, Einstein tensor, Einstein field equations
Changed3 November 2026

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